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A239098 Triangle read by rows: T(0,0)=1; T(m,0)=0; otherwise T(m,n) = (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1).

%I A239098 #12 Jul 30 2025 11:16:25
%S A239098 1,0,1,0,1,3,0,2,10,15,0,6,40,105,105,0,24,196,700,1260,945,0,120,
%T A239098 1148,5068,12600,17325,10395,0,720,7848,40740,126280,242550,270270,
%U A239098 135135,0,5040,61416,363660,1332100,3213210,5045040,4729725,2027025,0,40320,541728,3584856,15020720,43022980,85345260,113513400,91891800,34459425
%N A239098 Triangle read by rows: T(0,0)=1; T(m,0)=0; otherwise T(m,n) = (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1).
%C A239098 If the first column is omitted we get A075856, which has much more information about this triangle.
%D A239098 P. W. Shor, Problem 78-6: A combinatorial identity, in Problems and Solutions column, SIAM Review; problem in 20, p. 394 (1978); solution in 21, pp. 258-260 (1979).
%H A239098 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
%e A239098 Triangle begins:
%e A239098 1,
%e A239098 0, 1,
%e A239098 0, 1, 3,
%e A239098 0, 2, 10, 15,
%e A239098 0, 6, 40, 105, 105,
%e A239098 0, 24, 196, 700, 1260, 945,
%e A239098 0, 120, 1148, 5068, 12600, 17325, 10395,
%e A239098 0, 720, 7848, 40740, 126280, 242550, 270270, 135135,
%e A239098 ...
%p A239098 T:=proc(m,n) option remember;
%p A239098 if (m=0) and (n=0) then 1;
%p A239098 elif (m=0) or (n=0) then 0;
%p A239098 else (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1); fi; end;
%p A239098 M:=20;
%p A239098 for m from 0 to M do
%p A239098 lprint([seq(T(m,n),n=0..m)]); od:
%Y A239098 Cf. A075856.
%K A239098 nonn,tabl
%O A239098 0,6
%A A239098 _N. J. A. Sloane_, Mar 23 2014